Subgroups — C₅×#22275 #11944995

Order 1280.

938 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#492	C₂	1	1	yes	maximal, characteristic
2	C₅×#2328	C₂	1	1	yes	maximal, characteristic
3	C₅×#2327	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1961	C₂	1	1	yes	maximal, characteristic
5	C₅×#1960	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1880	C₂	1	4	yes	maximal, complemented
7	C₅×#1877	C₂	1	4	yes	maximal, complemented
8	C₅×#1835	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₅×SD₁₆	C₂²	1	4	yes	—
11	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×#127	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
14	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
15	C₅×#492	C₄	1	8	yes	complemented
16	C₂×C₅×#103	C₂²	1	4	yes	—
17	C₂×C₅×#102	C₂²	1	4	yes	—
18	C₂×C₅×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#391	C₂²	1	2	yes	—
21	C₅×#390	C₂²	1	2	yes	—
22	C₅×#387	—	2	4	no	—
23	C₅×#386	—	2	4	no	—
24	C₅×#382	C₂²	1	1	yes	characteristic
25	C₅×#382	C₂²	1	1	yes	characteristic
26	C₅×#376	C₂²	1	1	yes	characteristic
27	C₅×#340	C₂²	1	4	yes	—
28	C₅×#336	C₂²	1	2	yes	—
29	C₅×#336	C₂²	1	2	yes	—
30	#22275	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₅×Q₈	C₂³	1	2	yes	—
32	C₂²×C₅×D₄	—	2	2	no	—
33	C₂²×C₅×D₄	C₂³	1	2	yes	—
34	C₂³×C₄×C₅	—	2	2	no	—
35	C₂×C₅×SD₁₆	—	4	4	no	—
36	C₂×C₅×SD₁₆	—	4	4	no	—
37	C₂×C₅×SD₁₆	C₂×C₄	1	16	yes	complemented
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₅×C₄⋊Q₈	—	2	2	no	—
41	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
42	C₅×#127	—	2	2	no	—
43	C₅×#127	C₂×C₄	1	4	yes	—
44	C₄×C₅×Q₈	—	4	4	no	—
45	C₄×C₅×D₄	—	4	4	no	—
46	C₂×C₅×C₄⋊C₄	—	2	2	no	—
47	C₂×C₅×C₄⋊C₄	—	2	2	no	—
48	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
51	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
52	C₂×C₄²×C₅	—	2	2	no	—
53	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
56	C₅×#103	—	4	4	no	—
57	C₅×#102	—	4	4	no	—
58	C₅×#97	—	2	2	no	—
59	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
60	C₅×#95	—	2	2	no	—
61	C₅×#95	—	2	2	no	—
62	C₅×#95	—	2	4	no	—
63	C₅×#95	D₄	1	4	yes	—
64	C₂×#492	C₂×C₅	1	1	yes	characteristic
65	#2328	C₂×C₅	1	1	yes	characteristic
66	#2327	C₂×C₅	1	1	yes	characteristic, complemented
67	#1961	C₂×C₅	1	1	yes	characteristic
68	#1960	C₂×C₅	1	1	yes	characteristic, complemented
69	#1880	C₂×C₅	1	4	yes	complemented
70	#1877	C₂×C₅	1	4	yes	complemented
71	#1835	C₂×C₅	1	1	yes	characteristic, complemented
72	#1746	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂⁴×C₅	—	2	2	no	—
74	C₂×C₅×Q₈	—	2	2	no	—
75	C₂×C₅×Q₈	—	2	4	no	—
76	C₂×C₅×Q₈	—	2	4	no	—
77	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
78	C₂×C₅×D₄	—	2	8	no	—
79	C₂×C₅×D₄	—	2	2	no	—
80	C₂×C₅×D₄	—	2	4	no	—
81	C₂×C₅×D₄	—	2	4	no	—
82	C₂×C₅×D₄	—	8	4	no	—
83	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
84	C₂²×C₄×C₅	—	2	2	no	—
85	C₂²×C₄×C₅	—	2	2	no	—
86	C₂²×C₄×C₅	—	2	2	no	—
87	C₂²×C₄×C₅	—	4	4	no	—
88	C₂²×C₄×C₅	—	8	4	no	—
89	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
93	C₅×SD₁₆	—	4	16	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
100	C₅×C₄⋊C₄	—	2	8	no	—
101	C₅×C₄⋊C₄	—	2	4	no	—
102	C₅×C₄⋊C₄	—	2	4	no	—
103	C₅×C₄⋊C₄	—	4	4	no	—
104	C₅×C₂²⋊C₄	—	4	4	no	—
105	C₅×C₂²⋊C₄	—	8	4	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	—	2	2	no	—
108	C₄²×C₅	—	4	4	no	—
109	C₄²×C₅	C₂²×C₄	1	2	yes	—
110	C₄²×C₅	#41	1	2	yes	—
111	C₂²×SD₁₆	C₂²×C₅	1	4	yes	—
112	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
113	C₂×#127	C₂²×C₅	1	1	yes	characteristic
114	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
115	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
116	#492	C₄×C₅	1	8	yes	complemented
117	C₂×#103	C₂²×C₅	1	4	yes	—
118	C₂×#102	C₂²×C₅	1	4	yes	—
119	C₂×#97	C₂²×C₅	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
121	#391	C₂²×C₅	1	2	yes	—
122	#390	C₂²×C₅	1	2	yes	—
123	#387	—	2	4	no	—
124	#386	—	2	4	no	—
125	#382	C₂²×C₅	1	1	yes	characteristic
126	#382	C₂²×C₅	1	1	yes	characteristic
127	#376	C₂²×C₅	1	1	yes	characteristic
128	#340	C₂²×C₅	1	4	yes	—
129	#336	C₂²×C₅	1	2	yes	—
130	#336	C₂²×C₅	1	2	yes	—
131	C₂³×C₅	—	2	4	no	—
132	C₂³×C₅	—	4	2	no	—
133	C₂³×C₅	—	4	2	no	—
134	C₂³×C₅	—	4	4	no	—
135	C₂³×C₅	—	4	4	no	—
136	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₅×Q₈	—	2	8	no	—
138	C₅×Q₈	—	4	4	no	—
139	C₅×D₄	—	2	8	no	—
140	C₅×D₄	—	4	4	no	—
141	C₅×D₄	—	8	8	no	—
142	C₂×C₄×C₅	—	2	4	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	2	4	no	—
148	C₂×C₄×C₅	—	4	2	no	—
149	C₂×C₄×C₅	—	4	2	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	—	4	2	no	—
152	C₂×C₄×C₅	—	4	4	no	—
153	C₂×C₄×C₅	—	4	4	no	—
154	C₂×C₄×C₅	—	8	8	no	—
155	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
156	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
159	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
160	C₅×C₈	—	2	4	no	—
161	C₅×C₈	—	4	2	no	—
162	C₂²×Q₈	C₂³×C₅	1	2	yes	—
163	C₂²×D₄	—	2	2	no	—
164	C₂²×D₄	C₂³×C₅	1	2	yes	—
165	C₂³×C₄	—	2	2	no	—
166	C₂×SD₁₆	—	4	4	no	—
167	C₂×SD₁₆	—	4	4	no	—
168	C₂×SD₁₆	C₂×C₄×C₅	1	16	yes	complemented
169	C₂²×C₈	C₂³×C₅	1	2	yes	—
170	C₂²×C₈	C₂³×C₅	1	2	yes	—
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	C₂×C₄×C₅	1	4	yes	complemented
173	#127	—	2	2	no	—
174	#127	C₂×C₄×C₅	1	4	yes	—
175	C₄×Q₈	—	4	4	no	—
176	C₄×D₄	—	4	4	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
180	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
181	C₂×C₂²⋊C₄	—	2	2	no	—
182	C₂×C₂²⋊C₄	—	2	4	no	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
187	#103	—	4	4	no	—
188	#102	—	4	4	no	—
189	#97	—	2	2	no	—
190	C₄×C₈	C₂×C₄×C₅	1	4	yes	complemented
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	#95	—	2	4	no	—
194	#95	C₅×D₄	1	4	yes	—
195	C₂²×C₅	—	4	4	no	—
196	C₂²×C₅	—	4	8	no	—
197	C₂²×C₅	—	8	4	no	—
198	C₂²×C₅	—	8	2	no	—
199	C₂²×C₅	—	8	2	no	—
200	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
201	C₂²×C₅	#535	1	1	yes	characteristic
202	C₂²×C₅	#535	1	1	yes	characteristic
203	C₂²×C₅	#577	1	4	yes	—
204	C₄×C₅	—	2	4	no	—
205	C₄×C₅	—	4	4	no	—
206	C₄×C₅	—	4	2	no	—
207	C₄×C₅	—	4	2	no	—
208	C₄×C₅	—	8	2	no	—
209	C₄×C₅	—	8	2	no	—
210	C₄×C₅	#518	1	2	yes	—
211	C₄×C₅	#540	1	2	yes	—
212	C₂⁴	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
217	C₂×D₄	—	2	8	no	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	2	4	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	8	4	no	—
222	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	8	4	no	—
228	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
229	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
230	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
231	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
232	SD₁₆	—	4	16	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₂²⋊C₄	—	8	4	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	4	no	—
248	C₄²	C₂²×C₄×C₅	1	2	yes	—
249	C₄²	C₅×#41	1	2	yes	—
250	C₂×C₅	—	8	4	no	—
251	C₂×C₅	#2328	1	1	yes	characteristic
252	C₂×C₅	#2949	1	2	yes	—
253	C₂×C₅	#3221	1	2	yes	—
254	C₂×C₅	#3222	1	2	yes	—
255	C₂³	—	2	4	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	4	no	—
259	C₂³	—	4	4	no	—
260	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
261	Q₈	—	2	8	no	—
262	Q₈	—	4	4	no	—
263	D₄	—	2	8	no	—
264	D₄	—	4	4	no	—
265	D₄	—	8	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	8	8	no	—
279	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
280	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
281	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
282	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
283	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
284	C₈	—	2	4	no	—
285	C₈	—	4	2	no	—
286	C₅	#22275	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²	—	4	4	no	—
288	C₂²	—	4	8	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	8	2	no	—
292	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
293	C₂²	C₅×#535	1	1	yes	characteristic
294	C₂²	C₅×#535	1	1	yes	characteristic
295	C₂²	C₅×#577	1	4	yes	—
296	C₄	—	2	4	no	—
297	C₄	—	4	4	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	2	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	C₅×#518	1	2	yes	—
303	C₄	C₅×#540	1	2	yes	—
304	C₂	—	8	4	no	—
305	C₂	C₅×#2328	1	1	yes	characteristic
306	C₂	C₅×#2949	1	2	yes	—
307	C₂	C₅×#3221	1	2	yes	—
308	C₂	C₅×#3222	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#376 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₅×#2327	C₂	semidirect
5. C₅×#1960	C₂	semidirect
6. C₅×#1880	C₂	semidirect
7. C₅×#1877	C₂	semidirect
8. C₅×#1835	C₂	semidirect
9. C₅×#1746	C₂	semidirect
15. C₅×#492	C₄	semidirect
30. #22275	C₅	direct (N × H)
37. C₂×C₅×SD₁₆	C₂×C₄	semidirect
41. C₅×C₄⋊Q₈	C₂×C₄	semidirect
59. C₄×C₅×C₈	C₂×C₄	semidirect
66. #2327	C₂×C₅	semidirect
68. #1960	C₂×C₅	semidirect
69. #1880	C₂×C₅	semidirect
70. #1877	C₂×C₅	semidirect
71. #1835	C₂×C₅	semidirect
72. #1746	C₂×C₅	semidirect
116. #492	C₄×C₅	semidirect
168. C₂×SD₁₆	C₂×C₄×C₅	semidirect
172. C₄⋊Q₈	C₂×C₄×C₅	semidirect
190. C₄×C₈	C₂×C₄×C₅	semidirect
286. C₅	#22275	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#492	2
C₅×#2328	2
C₅×#2327	2
C₅×#1961	2
C₅×#1960	2
C₅×#1880	2
C₅×#1877	2
C₅×#1835	2
C₅×#1746	2
#22275	5

Subgroups — C5×#22275 #11944995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22275 #11944995